The exterior Schwarzschild metric is strange in a Minkowski space - the Big Bang comes to the rescue

The Schwarzschild metric says that the radial coordinate r has for local observers stretched by a factor

       1 / sqrt(1 - r_s / r)

relative to the global Schwarzschild coordinates. There r_s is the Schwarzschild radius of the mass M which is at the origin of polar coordinates.

If r is large, then the above is approximately

       1 + 1/2 * r_s / r.

If we integrate the extra length

        1/2 * r_s / r

from, say, 100 r_s to the infinity, the integral diverges logarithmically.

That is, there is an infinite extra distance which we must stretch from the Minkowski space, if we embed just 1 kilogram of mass into it. It is strange if just 1 kg of mass would have such infinite consequences for the metric.

Making a rubber sheet model for the metric is hard, or impossible, if a small mass causes an infinite change in the spatial metric.

Fortunately, we do not live in a Minkowski space but in an expanding FLRW universe where the mass density is roughly constant at the scale of 1 billion light years.

We may consider the 1 kg mass as a local dense spot in the mass distribution of the universe. Let us assume that the whole universe is full of thin gas or dust, and the 1 kg mass was born from a collapse of a spherical cloud of some 10^26 cubic meters of that matter. Then we probably (?) can ignore the Schwarzschild geometry farther away than 3 * 10^8 meters from that 1 kg of mass.

In our rubber model in an earlier blog post, we assumed that the rubber sheet is firmly attached to a circular frame. We can take the radius of the frame as 3 * 10^8 meters in this particular case.

For the Sun, the radius of the frame would be 300 light years. That is still tiny, compared to the size of the visible universe.

We need to replace the rubber sheet model with a rubber balloon model. A dense mass deforms the balloon. The deformation energy is roughly the same as the gravitational binding energy of the dense mass.

The visible universe seems to have a flat spatial metric. We must take the balloon to be very large, so that its spatial curvature is not obvious? We would only see a small part of the huge expanding balloon.